Passivity is a property of engineering systems, most commonly encountered in
analog electronics and
control system
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial ...
s. Typically, analog designers use ''passivity'' to refer to incrementally passive components and systems, which are incapable of
power gain. In contrast, control systems engineers will use ''passivity'' to refer to thermodynamically passive ones, which consume, but do not produce, energy. As such, without context or a qualifier, the term ''passive'' is ambiguous.
An
electronic circuit
An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or Conductive trace, traces through which electric current can flow. It is a t ...
consisting entirely of passive components is called a passive circuit, and has the same properties as a passive component.
If a device is ''not'' passive, then it is an active device.
Thermodynamic passivity
In
control system
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial ...
s and circuit network theory, a passive component or circuit is one that consumes energy, but does not produce energy. Under this methodology,
voltage
Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
and
current source
A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.
A current source is the dual of a voltage source. The term ''current sink'' is sometimes used for sources fed ...
s are considered active, while
resistor
A resistor is a passive two-terminal electronic component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...
s,
capacitor
In electrical engineering, a capacitor is a device that stores electrical energy by accumulating electric charges on two closely spaced surfaces that are insulated from each other. The capacitor was originally known as the condenser, a term st ...
s,
inductor
An inductor, also called a coil, choke, or reactor, is a Passivity (engineering), passive two-terminal electronic component, electrical component that stores energy in a magnetic field when an electric current flows through it. An inductor typic ...
s,
transistor
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch electrical signals and electric power, power. It is one of the basic building blocks of modern electronics. It is composed of semicondu ...
s,
tunnel diode
A tunnel diode or Esaki diode is a type of semiconductor diode that has effectively " negative resistance" due to the quantum mechanical effect called tunneling. It was invented in August 1957 by Leo Esaki and Yuriko Kurose when working ...
s,
metamaterial
A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is a type of material engineered to have a property, typically rarely observed in naturally occu ...
s and other dissipative and energy-neutral components are considered passive. Circuit designers will sometimes refer to this class of components as dissipative, or thermodynamically passive.
While many books give definitions for passivity, many of these contain subtle errors in how initial conditions are treated and, occasionally, the definitions do not generalize to all types of nonlinear time-varying systems with memory. Below is a correct, formal definition, taken from Wyatt et al., which also explains the problems with many other definitions. Given an ''n''-
port
A port is a maritime facility comprising one or more wharves or loading areas, where ships load and discharge cargo and passengers. Although usually situated on a sea coast or estuary, ports can also be found far inland, such as Hamburg, Manch ...
''R'' with a state representation ''S'', and initial state ''x'', define available energy ''E''
''A'' as:
:
where the notation sup
''x''→''T''≥0 indicates that the
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
is taken over all ''T'' ≥ 0 and all admissible pairs with the fixed initial state ''x'' (e.g., all voltage–current trajectories for a given initial condition of the system). A system is considered passive if ''E''
''A'' is finite for all initial states ''x''. Otherwise, the system is considered active. Roughly speaking, the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
is the instantaneous power (e.g., the product of voltage and current), and ''E''
''A'' is the upper bound on the integral of the instantaneous power (i.e., energy). This upper bound (taken over all ''T'' ≥ 0) is the ''available energy'' in the system for the particular initial condition ''x''. If, for all possible initial states of the system, the energy available is finite, then the system is called ''passive.'' If the available energy is finite, it is known to be non-negative, since any trajectory with voltage
gives an integral equal to zero, and the available energy is the supremum over all possible trajectories. Moreover, by definition, for any trajectory , the following inequality holds:
:
.
The existence of a non-negative function ''E''
''A'' that satisfies this inequality, known as a "storage function", is equivalent to passivity. For a given system with a known model, it is often easier to construct a storage function satisfying the differential inequality than directly computing the available energy, as taking the supremum on a collection of trajectories might require the use of
calculus of variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of f ...
.
Incremental passivity
In
circuit design
In electrical engineering, the process of circuit design can cover systems ranging from complex electronic systems down to the individual transistors within an integrated circuit. One person can often do the design process without needing a pl ...
, informally, passive components refer to ones that are not capable of
power gain; this means they cannot
amplify signals. Under this definition, passive components include
capacitor
In electrical engineering, a capacitor is a device that stores electrical energy by accumulating electric charges on two closely spaced surfaces that are insulated from each other. The capacitor was originally known as the condenser, a term st ...
s,
inductor
An inductor, also called a coil, choke, or reactor, is a Passivity (engineering), passive two-terminal electronic component, electrical component that stores energy in a magnetic field when an electric current flows through it. An inductor typic ...
s,
resistor
A resistor is a passive two-terminal electronic component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...
s,
diode
A diode is a two-Terminal (electronics), terminal electronic component that conducts electric current primarily in One-way traffic, one direction (asymmetric electrical conductance, conductance). It has low (ideally zero) Electrical resistance ...
s,
transformer
In electrical engineering, a transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple Electrical network, circuits. A varying current in any coil of the transformer produces ...
s, voltage sources, and current sources. They exclude devices like
transistor
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch electrical signals and electric power, power. It is one of the basic building blocks of modern electronics. It is composed of semicondu ...
s,
vacuum tube
A vacuum tube, electron tube, thermionic valve (British usage), or tube (North America) is a device that controls electric current flow in a high vacuum between electrodes to which an electric voltage, potential difference has been applied. It ...
s,
relay
A relay
Electromechanical relay schematic showing a control coil, four pairs of normally open and one pair of normally closed contacts
An automotive-style miniature relay with the dust cover taken off
A relay is an electrically operated switc ...
s, tunnel diodes, and
glow tubes.
To give other terminology, systems for which the
small-signal model is not passive are sometimes called locally active (e.g. transistors and tunnel diodes). Systems that can generate power about a time-variant unperturbed state are often called parametrically active (e.g. certain types of nonlinear capacitors).
Formally, for a memoryless two-terminal element, this means that the
current–voltage characteristic
A current–voltage characteristic or I–V curve (current–voltage curve) is a relationship, typically represented as a chart or graph, between the electric current through a circuit, device, or material, and the corresponding voltage, or p ...
is
monotonically increasing. For this reason, control systems and circuit network theorists refer to these devices as locally passive, incrementally passive, increasing, monotone increasing, or monotonic. It is not clear how this definition would be formalized to multiport devices with memory – as a practical matter, circuit designers use this term informally, so it may not be necessary to formalize it.
[This is probably formalized in one of the extensions to Duffin's Theorem. One of the extensions may state that if the small signal model is thermodynamically passive, under some conditions, the overall system will be incrementally passive, and therefore, stable. This needs to be verified.]
Other definitions of passivity
This term is used colloquially in a number of other contexts:
* A passive USB to PS/2 adapter consists of wires, and potentially resistors and similar passive (in both the incremental and thermodynamic sense) components. An active USB to PS/2 adapter consists of logic to translate signals (active in the incremental sense)
* A passive mixer consists of just resistors (incrementally passive), whereas an active mixer includes components capable of gain (active).
* In audio work one can also find both (incrementally) passive and active converters between balanced and unbalanced lines. A passive
balun converter is generally just a transformer along with, of course, the requisite connectors, while an active one typically consists of a differential drive or an instrumentation amplifier.
* In some books, devices that exhibit gain or a rectifying function (e.g.
diode
A diode is a two-Terminal (electronics), terminal electronic component that conducts electric current primarily in One-way traffic, one direction (asymmetric electrical conductance, conductance). It has low (ideally zero) Electrical resistance ...
s) are considered active. Only resistors, capacitors, inductors, transformers, and gyrators are considered passive.
United States Patent and Trademark Office
The United States Patent and Trademark Office (USPTO) is an List of federal agencies in the United States, agency in the United States Department of Commerce, U.S. Department of Commerce that serves as the national patent office and trademark ...
is amongst the organisations classing diodes as active devices. This definition is somewhat informal, as diodes can be considered non-linear resistors, and virtually all real-world devices exhibit some non-linearity.
* Sales/product catalogs will often use different informal definitions of this term, as fitting to a particular hierarchies of products being sold. It is not uncommon, for example, to list all silicon devices under "active devices," even if some of those devices are technically passive.
Stability
Passivity, in most cases, can be used to demonstrate that passive circuits will be stable under specific criteria. This only works if only one of the above definitions of passivity is used – if components from the two are mixed, the systems may be unstable under any criteria. In addition, passive circuits will not necessarily be stable under all stability criteria. For instance, a resonant series
LC circuit will have unbounded voltage output for a bounded voltage input, but will be stable in the sense of
Lyapunov, and given bounded energy input will have bounded energy output.
Passivity is frequently used in control systems to design stable control systems or to show stability in control systems. This is especially important in the design of large, complex control systems (e.g. stability of airplanes). Passivity is also used in some areas of circuit design, especially filter design.
Passive filter
A passive filter is a kind of
electronic filter
Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped-element model, lumped electronic components, as opposed to distributed-element filters. That ...
that is made only from passive components – in contrast to an active filter, it does not require an external power source (beyond the signal). Since most filters are linear, in most cases, passive filters are composed of just the four basic linear elements – resistors, capacitors, inductors, and transformers. More complex passive filters may involve nonlinear elements, or more complex linear elements, such as transmission lines.
A passive filter has several advantages over an
active filter
An active filter is a type of analog circuit implementing an electronic filter using active components, typically an amplifier. Amplifiers included in a filter design can be used to improve the cost, performance and predictability of a filter.
...
:
* Guaranteed stability
* Scale better to large signals (tens of amperes, hundreds of volts), where active devices are often expensive or impractical
* No power supply needed
* Often less expensive in discrete designs (unless large coils are required). Active filters tend to be less expensive in integrated designs.
* For linear filters, potentially greater linearity depending on components required (in many cases, active filters allow the use of more linear components; e.g. active components can permit the use of a polypropylene or NP0 ceramic capacitor, while a passive one might require an electrolytic).
They are commonly used in
speaker crossover design (due to the moderately large voltages and currents, and the lack of easy access to a power supply), filters in
power distribution
Electric power distribution is the final stage in the delivery of electricity. Electricity is carried from the transmission system to individual consumers. Distribution substations connect to the transmission system and lower the transmission v ...
networks (due to the large voltages and currents),
power supply
A power supply is an electrical device that supplies electric power to an electrical load. The main purpose of a power supply is to convert electric current from a source to the correct voltage, electric current, current, and frequency to power ...
bypassing (due to low cost, and in some cases, power requirements), as well as a variety of discrete and home brew circuits (for low-cost and simplicity). Passive filters are uncommon in
monolithic integrated circuit design, where active devices are inexpensive compared to resistors and capacitors, and inductors are prohibitively expensive. Passive filters are still found, however, in
hybrid integrated circuit
A hybrid integrated circuit (HIC), hybrid microcircuit, hybrid circuit or simply hybrid is a miniaturized electronic circuit constructed of individual devices, such as semiconductor devices (e.g. transistors, diodes or Integrated circuits, mo ...
s. Indeed, it may be the desire to incorporate a passive filter that leads the designer to use the hybrid format.
Energic and non-energic passive circuit elements
Passive circuit elements may be divided into energic and non-energic kinds. When current passes through it, an energic passive circuit element converts some of the energy supplied to it into
heat
In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
. It is
dissipative. When current passes through it, a non-energic passive circuit element converts none of the energy supplied to it into heat. It is non-dissipative. Resistors are energic. Ideal capacitors, inductors, transformers, and gyrators are non-energic.
[Nordholt, E.H. (1983). Design of High-Performance Negative Feedback Amplifiers, Elsevier Scientific Publishing Company, Amsterdam, , p. 15.]
Notes
References
Further reading
* —Very readable introductory discussion on passivity in control systems.
* —Good collection of passive stability theorems, but restricted to memoryless one-ports. Readable and formal.
* —Somewhat less readable than Chua, and more limited in scope and formality of theorems.
* —Gives a definition of passivity for multiports (in contrast to the above), but the overall discussion of passivity is quite limited.
*
— A pair of memos that have good discussions of passivity.
* —A complete exposition of dissipative systems, with emphasis on the celebrated KYP Lemma, and on Willems' dissipativity and its use in Control.
{{refend
Engineering concepts